Digital signals have become widespread and are favored in many applications because of their noise immunity. This is a result of their discreteness in both time and amplitude. Once the signal has been discretized, the signal can be stored or transmitted without additional noise being added. There are many applications however, where the discretization in both time and amplitude needs to be changed in the discrete domain. For example, if two digital systems operate at different sampling frequencies, a multirate system is used for the sampling rate conversion. The amplitude discretization can also be changed. Going from a 16 to an 8 bit representation, for example, would reduce the memory requirements for storing a signal. For instance, in gaming applications the precision of each sample is about 12 bits and the ring tones have a resolution of 8 bits. The process of lowering the amplitude resolution of a digital signal is called re-quantization.
If the signal amplitude change is large from sample to sample, then it is generally assumed that the re-quantization will be a uniformly (discrete) distributed i.i.d. (independent and identically distributed) sequence (white noise). In this case, the re-quantization error is independent of the signal being quantized. This assumption does not hold for all cases particularly when the signal has small amplitude. In this case rounding or truncating a signal can introduce various undesirable artifacts, namely, additional harmonics related to the signal being re-quantized. For this case, the re-quantization error is an autoregressive moving average (ARMA) process that cannot be modeled like a white noise sequence.
To avoid these unwanted harmonics, dither is generally added to the signal being quantized. Dither is an i.i.d. signal whose purpose is to ensure that the quantization error is uncorrelated with the signal being quantized. In addition, the dither signal is independent of the input. In additive dithering, the quantization error signal is dependent of the signal being quantized but some techniques can make the value of the first and second statistical moments independent of the error. The main disadvantage of adding dither is that since the dither is basically a noise signal, the signal to noise ratio (SNR) of the final re-quantized signal is lowered. Because of this, it is desirable to know if re-quantization will introduce the undesired harmonics. In some cases, the quantization noise will be an i.i.d signal even though no dither is added. In these cases, the signal can be re-quantized with no added harmonics, and without the signal to noise ratio penalty.
Another problem in the classic dithering model is that the amount of error is difficult to control when the probability density function (PDF) of the dither is uniform (RPDF) or gaussian (GPDF). It has been proved that dither with a triangular PDF (TPDF) can produce an error signal which has a constant variance, as this renders the first and second moments independent of the input. This dither has a larger variance in contrast to RPDF dither and the advantage of this dither is supported with psycho-acoustic tests which show that users prefer a constant noise instead of a non-constant noise. It has also been proved that, it is not possible to make a classic dither with a constant and lower variance in the quantization error than the one obtained with TPDF dither.
History of Quantization and Dithering
A lot of research has been done in the area of quantization and dithering. Different schemes of analog to digital conversion such as uniform and non-uniform quantization have been developed to sample continuous signals. For the purpose of this explanation, the quantization and re-quantization are assumed to be uniform where the difference between quantization levels is constant.
The study of quantization properties and its effects increased after the middle of the 20th century. An important mathematical foundation was published and recently summarized by Widrow (incorporated herein by reference). He establishes that quantization error can be modeled as a uniform i.i.d sequence under a given set of conditions. He found that the process of obtaining the probability density function (PDF) of the quantized signal is similar to Shannon's Sampling Theorem and named “Area Sampling”. It is explained how, having the relationship between the input and the output, and the PDF of the output of the quantizer, the original distribution could be recovered. Moreover, Widrow's research work defined the higher order statistics of quantization error and the study of the quantization output moments.
Dithering
The word dithering was originally used during in the Second World War. Aircraft bomb trajectory was more accurate when the airplane was flying, since the vibration reduces the error of moving parts. This vibration was termed dithering. Some initial applications of dithering were introduced by Roberts in his PhD thesis at MIT in (incorporated herein by reference). His work was related to the transmission of images in a digital television channel. To transmit a picture, the length of each sample was of at least 6 bits in the Pulse Code Modulation (PCM) standard. This is because the human eye is sensitive to the small changes in intensity. With the use of dithering, Roberts could reduce the resolution to 3 bits. In this work, he added small amounts of noise to the signal before quantization to cause the same effect in the perception of the eye with fewer bits. Some examples extracted from his thesis are shown in FIG. 1.
Later in 1964, Schuchman determined sufficient and necessary conditions of dither to have a minimum loss of statistical properties. For instance, one condition is that statistical dither must be independent of the signal to be quantized.
The state of the art dithering techniques used in industry about dithering were developed at least 20 years ago. Gray and Stokham summarized the most important research about dithered quantizers in (incorporated herein by reference). In this work, the theory of the subtractive and non-subtractive dithering and its statistical properties was explained. Also, Gray analyzed the spectra of quantization noise and summarized principal concepts about quantization, vector quantization, and dithering]. In addition, a strong mathematical foundation for the theory of non-subtractive dithering was developed in the AudioLab at University of Waterloo. These include the first and second order statistics of the system input and output and the introduction of digital dither.
In the middle of the 1980's, digital processing became more widespread and it was necessary to include dither in digital systems. In digital systems, the process of lowering the resolution of a signal is called re-quantization. Another study of digital quantization comes to the same conclusion as with continuous quantization. This work also introduces digital dither. Their work was extended when triangular PDF dither was used in digital audio. Currently, TPDF is the most popular technique of dithering in one-dimensional signals.
Dither is used in various applications. One of the most famous applications was developed in Bell Labs where Jayant and Rabiner used RPDF in speech processing (incorporated herein by reference). Another application uses GPDF in a high speed digital system for the suppression of the electromagnetic field. Furthermore, dithering is used in feedback systems to reduce the oscillation at high frequency.
Noise Shaping
One of the main objectives of this invention is to have white re-quantization error when quantizing signals. On the other hand, in some applications it is better to have a non-white noise. For example, human beings have greater perception of sound near to 4 kHz. It is possible to modulate (i.e, change the frequency) of the error signal to frequencies where we are less sensitive. In other words, this process changes the shape of the error spectrum to be minimally audible as it is shown in FIG. 3. This process is known as “noise shaping”. FIG. 2 illustrates a general scheme for noise shaping, where Q is the quantizer (re-quantizer) and H(z) is a feedback filter. The difference between the input to the quantizer and the output is filtered and added to the input to modulate the total error.
Knowing the advantages and disadvantages of the prior art, what is needed is the development and software/hardware implementation of an algorithm that measure the need for dithering and to develop a segment dependent dither where dither is added only to the segments where need dither, thus, providing adaptive dither with lower variance than the prior art.